Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.575646 
Title:  Set theoretic and topological characterisations of ordered sets  
Author:  Papadopoulos, Kyriakos B. 
ISNI:
0000 0004 2738 2029


Awarding Body:  University of Birmingham  
Current Institution:  University of Birmingham  
Date of Award:  2013  
Availability of Full Text: 


Abstract:  
Van Dalen and Wattel have shown that a space is LOTS (linearly orderable topological space) if and only if it has a T\(_1\)separating subbase consisting of two interlocking nests. Given a collection of subsets \(\mathcal L\) of a set X, van Dalen and Wattel define an order \(\triangleleft\)\(_\mathcal L\) by declaring \(_\mathcal X\) \(\triangleleft\)\(_\mathcal L\) \(_\mathcal Y\) if and only if there exists some L \(\in\) \(\mathcal L\) such that x \(\in\) L but y \(\notin\) L. We examine \(\triangleleft\)\(_\mathcal L\) in the light of van Dalen and Wattel’s theorem. We go on to give a topological characterisation of ordinal spaces, including \(_\mathcal W\)\(_1\), in these terms, by first observing that the T\(_1\)separating union of more than two nests generates spaces that are not of high ordertheoretic interest. In particular, we give an example of a countable space X, with three nests \(\mathcal L\),\(\mathcal R\),\(\mathcal P\), each T\(_0\)separating X, such that their union T\(_1\)separates X, but does not T\(_2\)separate X. We then characterise ordinals in purely topological terms, using neighbourhood assignments, with no mention of nest or of order. We finally introduce a conjecture on the characterisation of ordinals via selections, which may lead into a new external characterisation.


Supervisor:  Not available  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.575646  DOI:  Not available  
Keywords:  QA Mathematics  
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